The present invention relates to nonlinear dynamical systems, more particularly, to a system and method for filtering signals from a nonlinear dynamical system.
A deterministic signal from a nonlinear system may look like noise when displayed in either a time or frequency domain. Numerous work in this field involved a search for applications of these xe2x80x9cnoise-likexe2x80x9d deterministic signals. Cuomo and Oppenheim have applied a chaotic system with self-synchronization property to the secure communications problem (K. M. Cuomo and A. V. Oppenheim, xe2x80x9cChaotic Signals and Systems for Communicationsxe2x80x9d, IEEE Proceedings of International Conference on Acoustics, Speech, and Signal Processing, vol. 3, pp. 137-140, 1993). In their implementation, if a level of additive noise due to a transmitting channel exceeds 10% of the driving signal, synchronization will not occur. Therefore, for their algorithm to succeed at low signal-to-noise ratios (SNRs), a noise reduction algorithm is necessary.
Unfortunately, conventional linear filtering methods cannot be applied successfully to signals produced by chaotic systems, because the signals have, generally, broad-band spectra. Among the existing noise reduction methods, a method described in the article by J. D. Farmer and J. J. Sidorwich, xe2x80x9cOptimal Shadowing and Noise Reductionxe2x80x9d, Physical D 47, pp. 373-392, 1991 (xe2x80x9cFarmer""s methodxe2x80x9d) exhibits good performance in mild SNR circumstances. The Farmer""s method assumes the system dynamics are known. However, the structure of the Farmer""s method is relatively complicated because it combines the manifold decomposition procedure and singular value decomposition for the inversion of a large rank deficient matrix.
It is therefore an object of the present invention to provide generalized iterative noise reduction methods and systems for contaminated chaotic signals which are simple and easily implemented.
To attain the above object, according to an aspect of the present invention, there is provided a method and system for filtering signals from a nonlinear dynamical system, comprising the steps of: (a) setting an initial enhanced point to a noisy point; (b) estimating a deviation term; (c ) weighting the estimated deviation term; (d) computing a new enhanced point; and (e) iterating the steps (b) to (d) until the computed point converges to a true enhanced point.
According to another aspect of this invention, there is provided a method and system for filtering signals from a nonlinear dynamical system, comprising the steps of: (a) setting an initial enhanced point to a noisy point; (b) estimating an intermediate enhanced point; (c) computing a new enhanced point using the estimated point and weighting constants; and (d) iterating the steps (b) and (c ) until the computed point converges to a true enhanced point.
A system for filtering signals from a nonlinear dynamical system, includes means for setting an initial enhanced point to a noisy point of an input signal, means for estimating a deviation term, means for weighting the estimated deviation term and means for computing a new enhanced point wherein the means for computing iterates until the computed point converges to a true enhanced point.
Another system for filtering signals from a nonlinear dynamical system includes means for setting an initial enhanced point to a noisy point, means for estimating an intermediate enhanced point and means for computing a new enhanced point using the estimated point and a weighting constant, the computing means for iterating until the computed point converges to a true enhanced point.
These and other objects, features and advantages of the present invention will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings.